Design and Control of Compliant Tensegrity Robots Through Simulation and Hardware Validation

1,2 1,3 1,4 1,5 KEN CALUWAERTS ,JÉRÉMIE DESPRAZ ,ATIL ISÇEN¸ ,ANDREW P. S ABELHAUS , 1,6 2 1,7 JONATHAN BRUCE ,BENJAMIN SCHRAUWEN , AND VYTAS SUNSPIRAL

1Dynamic Tensegrity Robotics Lab, NASA Ames Research Center, USA 2Reservoir Lab, Electronics and Information Systems department, Ghent University, Belgium [email protected], [email protected] 3Biorobotics Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland jeremie.despraz@epﬂ.ch 4Oregon State University, USA [email protected] 5Berkeley Institute of Design, University of California Berkeley, USA [email protected] 6USRA/University of California Santa Cruz, USA [email protected] 7SGT Inc., NASA Ames Intelligent Robotics Group, USA [email protected]

Abstract

To better understand the role of tensegrity structures in biological systems and their application to robotics, the Dynamic Tensegrity Robotics Lab at NASA Ames Research Center has developed and validated two software environments for the analysis, simulation, and design of tensegrity robots. These tools, along with new control methodologies and the modular hardware components developed to validate them, are presented as a system for the design of actuated tensegrity structures. As evidenced from their appearance in many biological systems, tensegrity ("tensile-integrity") structures have unique physical properties that make them ideal for interaction with uncertain environments. Yet, these characteristics make design and control of bio-inspired tensegrity robots extremely challenging. This work presents the progress our tools have made in tackling the design and control challenges of spherical tensegrity structures. We focus on this shape since it lends itself to rolling locomotion. The results of our analyses include multiple novel control approaches for mobility and terrain interaction of spherical tensegrity structures that have been tested in simulation. A hardware prototype of a spherical six-bar tensegrity, the Reservoir Compliant Tensegrity Robot (ReCTeR), is used to empirically validate the accuracy of simulation.

Keywords: tensegrity, bio-inspired locomotion, central pattern generators, reservoir computing, compliant robotics, soft robotics, planetary exploration

This is a pre-copyedited, author-produced PDF of an article accepted for publication in Journal of the Royal Society Interface. The deﬁnitive publisher-authenticated version is available online at: http://dx.doi.org/10.1098/rsif.2014.0520 or http://rsif.royalsocietypublishing.org/content/11/98/20140520.

Please cite this work as: Caluwaerts K, Despraz J, Iscen A, Sabelhaus AP, Bruce J, Schrauwen B, SunSpiral V. Design and control of compliant tensegrity robots through simulation and hardware validation. J. R. Soc. Interface September 6, 2014; doi:10.1098/rsif.2014.0520 1742-5662

1 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

I. Introduction have no rigid connections between them [6]. This new view is challenging the "common sense" view of skeletal structures as Prior work has investigated the unique structural properties of the primary load-bearing elements of human and mammalian tensegrity systems, their role in biology, and control strategies bodies (see Figure 2). In the emerging "bio-tensegrity" model, for different tensegrity morphologies. One of the centers for bones are still under compression, but they are not passing com- this research is NASA Ames Research Center, where there is pressive loads to each other; rather, it is the continuous tension interest in these systems for planetary exploration missions. network of fascia that are the primary load-bearers [5, 6].

I.1 Tensegrity Structures Tensegrity structures are composed of compression elements encompassed within a network of tensional elements; consequently, each element experiences either pure compression or pure tension. This allows individual elements to be extremely lightweight, as designs do not need to resist bending or shear forces. Active motion in tensegrity structures can be performed with minimal energy expenditure since actuators work linearly along load paths in tension elements, avoiding torques caused by long lever arms of traditional robotic designs. A unique property of tensegrity structures is how they inter- nally distribute forces. Since there are no lever arms, forces do not magnify around joints or other common points of failure. Rather, externally-applied forces distribute through the Figure 2: Tensegrity models of the spine show how vertebrae structure via multiple load paths, creating system-level me- float without touching. Image courtesy of Tom Flemons (copy- chanical robustness and tolerance to forces applied from any right 2006) [7] direction or failure of individual actuation elements [1]. Thus, tensegrity structures are ideally suited for operation in dynamic environments where contact forces cannot always be predicted. I.3 Tensegrity Robotics for Space Exploration NASA is supporting research into tensegrity robotics to create I.2 Tensegrity and Biology planetary rovers with many of the same qualities that bene- fit biological systems. The high strength-to-weight ratio of tensegrity structures is attractive due to the impact of mass on mission launch costs. Likewise, large tensegrity structures are deployable from compact configurations, enabling them to fit into space-constrained spacecraft. While these qualities have inspired studies of deployable antennae and other large Figure 1: Computer simulations of a nucleated tensegrity cell space structures [8], the unique force distribution of tensegrity model exhibits mechanical coupling between the cell, the cy- robots has only recently been investigated for planetary ex- toskeleton, and the nucleus. Adapted by permission from ploration [9]. Initial work in the NASA Innovative Advanced Macmillan Publishers Ltd: Nature Reviews Molecular Cell Concepts (NIAC) project [10] shows that controllable compli- Biology [2], 2009 ance and force distribution properties allow for reliable and robust environmental interactions during landing and planetary Tensegrity structures are being discovered in many aspects surface exploration. of biological systems, which motivates this work’s bio-inspired A key goal of this NASA work is to develop a tensegrity modeling and controls approaches. The tensegrity concept ap- probe with an actively controlled tensile network, enabling pears at various scales, from the cytoskeleton of individual cells compact stowage for launch followed by deployment for land- (see Figure 1) [2, 3, 4] to mammalian physiology [5]. Emerg- ing. Compliant tensegrity probes can safely absorb significant ing biomechanical theories are shifting focus from bone-centric impact forces, enabling high-speed Entry, Descent, and Land- models to fascia-centric models. Fascia is the connective tissue ing (EDL) scenarios where the probe acts like an airbag [9]. in our bodies (including muscles, ligaments, tendons, etc.) that However, unlike an airbag that must be discarded after a single forms a continuous web of tension, even surrounding and sup- use, the tensegrity also provides rolling mobility (Figure 3). porting bones, which, unlike traditional mechanical systems, This enables compact and lightweight planetary exploration

2 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

trol algorithms based on Central Pattern Generators (CPGs), distributed learning, and genetic algorithms instead of more traditional control approaches.

I.4.1 Central Pattern Generators CPGs are neural circuits found in both vertebrate and inverte- brate animals that produce rhythmic patterns of neural activity without receiving rhythmic inputs. CPGs have been studied from a biological perspective and have been used extensively in robotics [21]. The term central indicates that rhythms of the CPG are not driven by sensory input, but are self-generated. CPGs are the fundamental building blocks for locomotor neural circuits in many animals, and are also key to other fundamental rhythmic activities such as chewing, breathing, and digesting. Recent research shows a close relationship between CPGs and motion primitives in the spine that enable both rhythmic and Figure 3: Mission Scenario - A tightly packed set of tensegri- discrete motions [22]. Alongside their biological inspiration ties expands, spreads out, falls to surface of moon, and then for use in robotic motion, CPGs present several interesting and safely bounces on impact. The same tensegrity structure cush- useful properties including distributed control, robustness to ioning landing is then used for exploration. perturbations, inherent tolerance to redundancies, fast control loops, and the ability to modulate locomotion by simple control missions with the capabilities of traditional wheeled rovers, signals. but with a mass and cost similar to a stationary probe. Dual CPGs are therefore well suited for controlling tensegri- use of structure allows a tensegrity mission to have a high ties [23] and other biomimetic structures [24]. Additionally, mass fraction between science payload and overall weight (as there is intuition for pairing tensegrity robots with CPG net- measured at atmospheric entry). This reduces mission cost works: the dynamics of physical forces propagating through and enables new forms of surface exploration utilizing the a tensegrity structure are similar to the dynamics of control tensegrity’s natural tolerance to impacts [9]. patterns propagating through CPG networks.

I.4 Tensegrity Control I.4.2 Evolutionary Algorithms Instead of defining a control policy directly, evolutionary al- Tensegrity structures are a fairly modern concept having been gorithms can be used to learn a control policy. This is accom- initially explored in the 1960s by Buckminster Fuller [11] and plished through an iterative cycle, where in simulation a control the artist Kenneth Snelson [12]. Initial tensegrity research was policy is run and evaluated, and this evaluation is fed back into mostly concerned with form-finding techniques [13] and the the genetic algorithm so that it can improve the control policy. design and analysis of static structures [14, 15]. Research into Evolutionary algorithms have the following advantages: control of tensegrity structures began in the mid-1990s, with initial efforts at formalizing the dynamics of tensegrity struc- 1. Complex, nonlinear control policies can be learned. tures only recently emerging [15]. The very properties that make tensegrities ideal for physical interaction with the envi- 2. Underlying dynamics need not be known. ronment (compliance, multi-path load distribution, nonlinear 3. Control policies can be learned or parameters of existing dynamics, etc.) also present significant challenges to tradi- control policies can be optimized. tional control approaches. A recent review shows that there are still problems actively controlling tensegrities [16]. Work has 4. Distributed learning can be used to scale to larger tenseg- continued in the analytical understanding of the equations of rities and to accelerate learning. motion and dynamics of tensegrity structures [17]. However, environmental interactions cause additional modeling difficul- Moreover, coevolutionary algorithms provide distributed ties, typically limiting the effectiveness of such approaches. learning for multiagent problems [25]. Each component can Consequently, successful demonstrations of tensegrity mobil- individually learn a control policy that decides how to actuate ity have primarily used non-analytical methods [1, 18, 19, 20]. its individual end cap in such a way that global performance Our approach to these problems is therefore to develop con- is maximized [26]. Challenges controlling tensegrity robots

3 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface using traditional approaches have led some researchers to con- However, its simple underlying model makes it unsuited for sider biologically-inspired evolutionary or coevolutionary al- the study of complex environmental interactions. gorithms [1, 27, 28, 29]. II.3 NASA Tensegrity Robotics Toolkit I.5 Outline Our main simulation environment, the NASA Tensegrity This paper is organized as follows: section II introduces our Robotics Toolkit (NTRT), is built on the discrete time Bul- simulators and hardware platform; section III presents our let Physics engine (a game physics simulator) [30]. simulator validation results; section IV describes our various Since game physics require real-time simulation, Bullet is control strategies; and section V discusses our future work and designed to handle collisions without excessive processing our results and control strategies in context of other work. And power. However, the Bullet physics library does not currently last, section VI presents our conclusions. provide models of ropes, cables, or springs with realistic material properties and stress analysis. Instead of using these default soft body models, we built an additional library to sim- II. Systems and Models ulate spring-cable assemblies as two-point tensional elements that apply directional forces to rigid bodies. We introduce three systems to evaluate the design and con- This approach gives the ability to calculate the amount of trol of tensegrity structures. The ﬁrst two are simulation stretch and tension for each simulated cable, as well as the force environments, while the third is an untethered, lightweight exerted to the bodies, using more mathematically-rigorous robot prototype. Our simulators and hardware designs models. A current limitation of this method is that the cables do are Open Source and can be obtained from our website: not exist in the simulation world as physical bodies. Thus, their http://ti.arc.nasa.gov/tech/asr/intelligent-robotics/tensegrity/. collisions and interaction with rigid bodies are not simulated. For locomotion tasks and terrain types we have explored so II.1 Spring-Cable Assemblies far, this is not a problem, but will be addressed in the future to simulate more extreme terrain interactions. All tensile members of the structures we study in this work are compliant and we refer to them as spring-cable assemblies. II.4 Prototype Although various implementations are possible, all spring- cable assemblies in this work can be modeled as a zero rest- In addition to these two software environments, a physical length passive spring in series with a non-elastic cable. The prototype of the six-bar tensegrity was constructed. ReCTeR is tension on spring-cable assembly i is given by: a highly compliant, lightweight (1.1kg), underactuated tensegrity icosahedron robot, as shown in Figure 4. The robot’s 24 = max(p − p − ,0), fi ki i,0 i,1 i (1) p − p where ki is the spring stiffness, i,0 i,1 is the Euclidean distance between the attachments of the spring-cable assembly, and i is the cable length. Actuated members have a controllable rest length i.

II.2 Euler-Lagrange Simulator We extended the Euler-Lagrange formulation described in Skel- ton’s reference work on tensegrity systems by adding support for ground contacts and gravity [15]. In Skelton’s work, the tensegrity structure struts are modeled as cylinders with in- ﬁnitesimal radius. Strut-to-strut contacts are not modeled, Figure 4: ReCTeR: an untethered, highly compliant, spherical which is an acceptable approximation for NASA-scale mis- tensegrity robot. Top left: deployed robot. Center right: active sions. folding. Bottom: ReCTeR rolling from right to left. We found that this environment provides particularly accu- rate results for two types of experiments: tests of structural outer tensile elements are passive spring-cable assemblies with forces, and tests of effective structural stiffness. This simula- low stiffness springs (28.4N/m) under moderate pretension (≈ tor is used for payload acceleration prediction during impacts 10N). Six actuated spring-cable assemblies run through the from drop tests, as well as stiffness analyses and form-ﬁnding. robot connecting active and passive end caps (also see section

4 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

II.5). Their rest length is adjusted by a rotational DC mo- tensegrity icosahedron tensegrity icosahedron ReCTeR with payload tor (4.5W, 4.4:1 gear ratio) that spins a snag-resistant bobbin (5.5mm diameter). The other end of the cable attaches to a stiffer spring (81N/m) affixed to a passive end cap. We use low stiffness springs to allow active folding (Figure 4) without plas- tic deformation of the 24 passive tensile members or excessive 1m motor power requirements. shell spring-cable assemblies struts payload spring-cable assemblies payload The six active spring-cables run through the robot and con- ReCTeR actuated spring-cable assemblies nect non-parallel struts in an advantageous way. Stiffness analysis revealed that this pattern allows for large shape de- Figure 5: The various tensegrity configurations used in this formations with low motor power requirements. As a conse- paper. Left: tensegrity icosahedron with only outer shell mem- quence of low spring stiffnesses, the lowest natural frequencies bers. Center: tensegrity icosahedron with a payload by inner of oscillatory modes for the structure are on the order of a few elements. Right: ReCTeR configuration with passive outer Hz. shell and actuated spring-cable assemblies. Sensing and feedback control are achieved by 24 tension sensors using strain gages, six ground reaction force sensors, and three six-DOF inertial measurement units distributed evenly III. Validation of Simulations among the actuated end caps. To allow dynamic motion and rolling, each self-contained strut holds a hardware module with III.1 Experimental Setup battery power and wireless communication. The battery is To track the full state of the robot, an active marker motion mounted in the center of the strut to minimize the moment of capture setup was used. Passive struts were fitted with two inertia around its longitudinal axis. This makes ReCTeR fully markers, active struts received three markers. untethered.

III.2 Kinematics II.5 Robot Models The forward kinematics of the Euler-Lagrange and NTRT simulators were compared against motion capture data from Our control methods are implemented on platforms with vary- ReCTeR. ing configurations. Figure 5 shows the three tensegrity icosa- The six-strut ReCTeR robot was placed on one of its trian- hedra analyzed in this paper. gular faces and two top spring-cable assemblies were actuated, We put a particular emphasis on spherical icosahedron as shown in Figure 6. We tracked vertical displacement of an tensegrities. This symmetric configuration provides a large end cap not directly actuated by one these two members. The interior volume with a moderate number of members and can incident strut was suspended in air by a total of ten springs. be folded easily. It lends itself naturally to rolling locomotion Lengths of the two actuated spring assemblies were var- because of its triangular faces. Additionally, tensile-member ied from the point of no tension in the given configuration failure will result in reduced locomotion capabilities instead (slack) to 0.32m beyond this length. Each range was sampled of full failure, due to its redundant number of tensile mem- at ten equally spaced lengths, resulting in 100 measurement bers [31]. positions. Ranges were manually tuned to maximally deform the robot without causing it to roll. This experiment was re- The basic tensegrity icosahedron is shown on the left. This peated three times with no meaningful difference in observed structure has 24 spring-cable assemblies and six rigid rods. displacements. The spring-cable assemblies are also referred to as outer shell The average observed difference between motion capture elements. data and the Euler-Lagrange simulator was 6.5mm. For NTRT, A tensegrity icosahedron with a payload is displayed in we obtained an average error of 15mm (0.5% and 1.3% of the center. This structure has an additional 12 spring-cable ReCTeR’s diameter). assemblies to suspend the payload in the center of the robot. We also refer to these as inner members. III.3 Dynamics The model on the right displays ReCTeR’s configuration with actual dimensions. ReCTeR has the basic tensegrity icosa- Next, we compared dynamics of the NTRT simulator to hedron configuration with six extra actuated spring-cable as- ReCTeR hardware. This experiment was designed for two semblies. purposes: to first verify that the simulator can replicate ground

5 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

Figure 6: Kinematic comparison of Euler-Lagrange and NTRT simulators and ReCTeR motion capture data. The top left plot shows the experimental setup. The rest length of two actuated spring-cable assemblies (dashed lines) is modiﬁed. The full range of tracked end-cap motion during the experiment is shown in light yellow (convex hull). The end caps indicated by small black squares are on the ground. The three other plots show vertical displacement of the end cap indicated by the large black dot in the top left plot as a function of the lengths of the two actuated cables. The end cap where we trace the displacement is not directly actuated and is ﬂoating. The nodal displacement as a function of the actuator position is nonlinear, even for modest displacements. Note that the left-most point (0.05,0.05,0) is the reference point; displacements are relative to this initial state.

6 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface ot b ro NTRT

Figure 7: Comparing the dynamics of the robot and NTRT. The tensioned spring-cable assembly indicated by the dashed line is released (0.32m to 0.535m at 0.6m/s), causing the robot to topple. Two other actuated members are also tensioned, while the other three actuated springs are at their initial lengths, resulting in two slack springs. We observed a time-averaged error of the end caps’ vertical positions of less than 5% of ReCTeR’s diameter for all end caps. interactions; and second, that it can simulate conversion of pendix A provides a summary of the control methods in this potential energy into kinematic energy when a spring is re- work and an overview of related work. leased. The experimental setup is shown in Figure 7. The robot initially has a non-minimal ground contact (four end caps on IV.1 Coevolutionary Control the ground instead of just three), and three actuated springs are tensioned. Next, one of the tensioned, actuated springs is The first control method from our group is based on coevolu- loosened by its actuator, causing the robot to roll over. tionary algorithms [25]. We demonstrated successful rolling Since the experiment also depends on ReCTeR’s initial state, locomotion of a tensegrity icosahedron with this technique in the observed initial state from the motion capture data was the NTRT [29]. In this scheme, each spring-cable assembly is copied to the NTRT simulator. The ReCTeR model in the active and has a controller that evolves independently from the NTRT simulator was then released from this initial configura- other controllers (i.e., in independent pools), but cooperation tion, allowing it to reach the simulated, predicted equilibrium. is used to optimize behavior of the entire robot. The objec- Recorded motor positions from the physical test were then tive function for this maximization was set to be ReCTeR’s applied into the simulator, causing a similar roll-over motion. distance traveled during a fixed amount of time. The simplest implementation of this technique is an optimization of open- loop control signals that are only a function of time; sinusoidal IV. Locomotion Control functions performed well. After this method was explored, the effects of different com- Once it had been determined that the NTRT simulator modeled plexities and frequencies of these open-loop signals were ana- these robotic dynamics reasonably well, locomotion experi- lyzed. More precisely, we optimized step-wise functions with ments were performed with tensegrity robots in both simulation varying numbers of via points. This enables the study of com- and hardware. The algorithms described in section IV.1 apply putational load and scaling properties needed to estimate power to various configurations, but are presented here in simulation consumption of various controllers, as well as to investigate for the first configuration, as shown in Figure 5 with 1.5 me- effects of actuator failure. Figure 8 shows the learning curve ter rods, weighing 15kg. The controls in section IV.2 were of this process. In this case, optimized rest length signals had applied to simulations of the second configuration. Section four via points. An analysis of practical aspects of these results IV.3 presents hardware results on ReCTeR and a comparison (power consumption, actuator failure, etc.) is under way. of those results to previously-simulated implementations. Ap- While these open-loop controllers demonstrated basic

7 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface rolling behavior, they commonly failed in the presence of external forces or unexpected terrain conditions. To solve this problem, we developed a simple rolling algorithm that uses ground contact sensors located on the simulated end caps. Preliminary results have shown steerable rolling on various terrains. This brief analysis of coevolutionary learning for icosahedral tensegrity locomotion demonstrates that learning-based controls can provide robust rolling locomotion without analytical knowledge of the robot’s dynamics.

) 35 0.05 m ( 30 0.04 d

e 25 Best Policy ll 0.03

o 20

R 15 Average Failures 0.02 10 0.01

5 Failure rate 0 stance 0 Di 0 2000 4000 6000 8000 10000 12000 Simulations Figure 9: Regular icosahedron tensegrity shape with central payload (see Figure 5 center). The triangular contact surface Figure 8: Distance covered by the robot in 60s with distributed with the ground, highlighted in green, creates a reaction force learning of open-loop controllers based on coevolutionary N that, at rest, balances the weight of the structure, represented algorithms. Each of the 24 outer shell spring-cable assembly on the ﬁgure by the red arrow mg. Torque is created on the controllers has a different evolution pool, but their combined whole structure when displacement of the center of mass from behavior is optimized. its rest position occurs.

IV.2 Bio-Inspired Control was computed as the average of the two end points’ height. In contrast to the direct learning technique presented above, We understand that omnidirectional distance sensors can be our second set of approaches is more designer-involved and difficult to realize in hardware; it is not practical to rely on the specific to this structure. State feedback was used to increase full state of the robot as input. However, multiple solutions to rolling performance of the tensegrity robot with payload simu- this problem exist. An interesting approach is to embed ground lated with the NTRT (Figure 5). reaction-force sensors in a protective soft cap on each rod end. The idea behind these control laws is to create torque by A second possibility, motivated by the separation principle of moving the robot’s center of mass with respect to ground con- control theory [32], is to estimate system states from other tact surface to cause the robot to roll, as illustrated in Figure 9. sources, such as accelerometer and gyroscope measurements. This motion is achieved with a two-layer control architecture: This second approach could be augmented with the knowledge the robot’s heading and speed are controlled by displacement that this icosahedron robot often rolls over an edge of a face of the central payload using the inner spring-cable assemblies, triangle [20]. Finally, one could use a CPG to emulate rhythmic and motion is simplified by actuating the outer shell. activation of sensors, similar to the approach in section IV.2.2. Three control approaches test inner spring-cable assemblies: The control for the outer-shell cables was designed to tighten reactive controls, Inverse Kinematics-based controls (IK), and the bottom part of the structure when rolling, changing the lever CPG-based controls. Outer spring-cable assemblies are con- arms of the gravitational force from the robot’s center of mass, trolled with hand-tuning. Actuation of the outer shell reduces requiring less force to induce a roll. Typically in the presence ground contacts, not directly influencing heading or speed. of a slope, reduction of ground contact surface is sufficient This affects motion in several ways. First, it allows for creation to cause a roll down the slope. In order to take this into ac- of greater torques with the same payload displacement. Second, count, we added a measure of speed, which is computed as it smoothens rolling behavior of the structure by preventing the dot product between the center-of-mass position and the discontinuities due to excessive ground contact. robot’s overall heading direction vector. With this method, For each control approach, inputs were taken as functions of speed is a scalar number and its sign depends on the robot’s the robotic state. The height of each strut was computed using heading(positive in the desired direction and negative other- simulated omnidirectional distance sensors located at the end wise). Speed can then be used as feedback to influence the of each rod. The height assigned to each spring-cable assembly spring actuator command. Rest lengths of the shell spring-

8 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface cable assemblies are computed using the following actuation rule: v v v 2 2 ˙ = + min( , ) − , ≥ 0 vv11’’ w 0 h h0 speed i s i i v1v1v1v1 (2) v2’’ ˙ ¯ v4 = − , v4v4v4 vv44’ ’ i ws i otherwise v2v2

v3’ v3 v3’ v3 where hi is the height of spring-cable assembly i as measured from the distance sensors; is its current rest length; 0, h0, ¯ i and are constant parameters; and ws ∈ R+ accounts for the time scale where length corrections occur. 0 and h0 rep- Figure 10: Computation of new rest lengths according to resent the offset-rest length of the spring and the maximum ¯ the spring-cable assemblies’ individual orientations vi (time height measurement. The parameter represents the default ( −1) t n ). Length modification is indicated by colored lines, rest lengths of the springs that, if given as a command to all dashed red if reduced and green if elongated. The resulting motors, puts the tensegrity in a stable position on the ground. effect is displacement of the central payload in the desired Input and output parameters of this control law are updated ( ) ( −1) direction v (time t n = t n + dt). continuously through feedback control. Impedance control, which was adapted to tensegrities previously [24, 33], is used to modify spring-cable rest lengths. IV.2.2 CPG Controls CPGs have been successfully used in past tensegrity sys- IV.2.1 Reactive Controls tems [23]. Such controls are a feasible alternative to reactive controllers that enable generation of regular motion pat- The first technique for actuation of inner payload spring-cable terns. For this control, full-state information is used to generate assemblies was the use of reactive controllers.We note that the smooth motion under reactive controls. Then, resulting peri- only controllable parameter is cable length. The variables i odic commands were stored as a stable limit cycle of a CPG. here are the rest length of the inner springs. Global heading Once this process completes, the tensegrity can be driven by direction in a chosen inertial reference frame is defined by the CPG output with much less feedback. We used an adaptive unit vector v and the orientation of each spring in this same frequency Hopf oscillator [34] during the learning phase where reference frame, represented by the vector v . For each inner i the tensegrity was reactively control-driven. The underlying spring-cable assembly, we use the dot product d = v · v as i i dynamical system reads: feedback to control the position of the payload as follows: 2 2 u˙ = γ(μ − (u + g ))u − ωg + b(t) (5) ˙ =( + γ −p − p ) i 0 di i,0 i,1 wr (3) 2 2 g˙ = γ(μ − (u + g ))g + ωu (6) (0)=0 (4) i g ω˙ = −b(t) (7) u2 + g2 where the weight wr determines reactivity of the system and γ < 0 is a fixed parameter. Thus, without any external pertur- where u, g, and ω are state variables of the dynamical system, + γ bation, the system has a stable equilibrium position at 0 di . γ is a time constant, μ is the target frequency, and ω the target Rest length of the spring-cable assemblies where the orienta- pulsation of the signal. Note the element and time indices ( ) tion aligns with the global heading is reduced. Vice-versa, are dropped to simplify notation: u designates ui t , with springs pointing in the opposite direction are elongated. The i the index of the spring-cable assembly. The output u can global result is displacement of the payload in the direction synchronize to any periodic input signal b(t) and replaces the of the heading vector, as shown in Figure 10. Note that the feedback signal from the previous section. heading direction v can be chosen arbitrarily and adjusted dy- ( )= ( ). namically. This method resulted in stable and smooth rolling di t ui t (8) gaits, allowing a roll of up to 1m/s (≈1 body-length per sec- Once the signal is learned, time dependency of the pulsation ond) over flat terrain. The robot could also handle slopes up to is removed, i.e., ω is held constant and a term accounting for 8◦, bumpy terrain, obstacles, and collisions. ground-contact coupling is added to the dynamical system: The main disadvantage of the reactive method is the type 2 2 and amount of sensor feedback required to implement this u˙ = γ(μ − (u + g ))u − ωg − ηh(t) (9) approach in hardware. This issue is addressed by the control 2 2 g˙ = γ(μ − (u + g ))g + ωu (10) methods presented next, which are based on the same physical ω˙ = 0 principle but require less feedback information. (11)

9 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface where h(t) denotes the height signal fed back by the omnidi- reactive CPG Hybrid rectional ranging sensors and η ∈ R+ is a coefficient. This average speed [m/s] 1.00 0.50 0.38 method has the advantage of requiring minimal feedback and complex terrain Yes No No thus, minimal computations. However, it is important to note that the dynamical system runs on a much larger time scale Table 1: Bio-inspired control strategies summary than perturbations disturbing the system. A tensegrity driven only by a CPG would then, in the best case, only have a stable rolling gait on a flat, obstacle-free terrain. Consequently, it is pseudo-inverse of the Jacobian defined by: necessary to include a second control method that works on ∂ this smaller time scale and gives an appropriate response to = fi . J ∂(δ ) (16) j external perturbations. ij Note that this matrix is not the same as the one in the Taylor IV.2.3 Hybrid CPG - Inverse Kinematics Controls expansion of p(). The final control method we tested is a hybrid technique with The outputs of the IK algorithm ξ = δ∞ (where ∞ indi- inverse kinematics. First, the position of the central payload cates convergence) represent length corrections that have to p = px, py, pz is defined as a function of the inner cable be made to reposition the payload at the desired location. The lengths = (1, ..., n). We can write this as a small displace- outputs ξ can be used together with the adaptive frequency ment δp of the payload: oscillator as presented in section IV.2.2. This approach is inspired by two previous works by Ajallooeian et al. [35] and n ∂ ((0)) ξ δ ≈ ((0))+ pi δ Gay et al. [36]. We update if and only if the payload position pi pi ∑ ∂ j j=1 j lies on the opposite side of the robot’s center of mass, and ξ 2 (0) we continuously adjust with time according to the following 1 n n ∂ p ( ) + ∑ ∑ i δ δ (12) evolution rule: 2 ∂ ∂ j k d j=1 k=1 j k ξ(t)=−αξ(t) (17) dt or with α ∈ R+. In this way, corrections are made only if the tensegrity can potentially roll in an undesired direction. Note δ ≈ ((0))+ (p(0))δ pi pi J also that in order to use this method, both the position of the i 1 (0) payload and the center of mass are required inputs. Combining + δT H p ( ) δ (13) 2 i a corrective term with the output of the oscillator, the resulting dynamical system reads: ∂2 pi ∈{, , } ( )= 2 2 for i x y z and where H pi ∂ ∂ is the ˙ = γ(μ − ( + ))( − ξ( )) − ω − η ( ) (18) j k u u g u t g h t jk 2 2 =[∂ /∂ ] ˙ = γ(μ − ( + )) + ω( − ξ( )), (19) Hessian matrix associated with pi and J pi lj ij. g u g g u t If the value of ξ( ) is constant over time, the dynamical system Considering Eq. 13, we additionally define Δp = δp − t (0) converges asymptotically to u(t)=ξ [36]. While the pure p( ) and f(δ; Δp) as CPG implementation does not allow any steering control, this 1 implementation enables guidance on a desired trajectory on (δ; Δp)= (p(0))δ + δT ((0)) δ − Δ . fi J H pi pi i 2 flat terrain (see Figure 11). (14) Table 1 provides a summary of results obtained with differ- Computation of the spring-cable rest length changes δ for a ent control strategies over regular, flat terrain. desired payload displacement Δp corresponds to finding the δ that cause f = 0. Since this last equation is over-determined, Note that results do not take the trajectory of the path into nonlinear, and might not possess a real solution, we employed a account and, consequently, even if the distance traveled using quasi-Newtonian iterative method to approximate the solution. the CPG controller without any trajectory control is larger than Starting from a candidate solution, e.g., δ0 = 0, the next with hybrid control, the "quality" of the path is worse (e.g., iteration is computed as: see Figure 11). Interestingly, we observe the stable gait pattern obtained in simulation is a sequence of contacts defined −1 δ +1 = δ − f(δ ; Δp) as energetically optimal by Koizumi et al. [20] for a tenseg- k k Jk k (15) rity icosahedron. With the current implementation, only the −1 until convergence. J here denotes the Moore-Penrose reactive controller manages to induce rolling efficiently over

10 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

Figure 11: Trajectory of the tensegrity (top view). The black curve represents the trajectory while the robot is driven by the reactive control algorithm and the CPG is in the learning mode (50s). Motion is regular and the heading is maintained throughout the entire period. Yellow and red trajectories represent the path traveled once the CPG controller takes over (40s). When the CPG is coupled to the height signal and receives inputs from the second-order inverse kinematics algorithm (red curve), the resulting trajectory is a long and relatively straight Figure 12: Examples of successful locomotion over complex line extending well the reactive control. terrains, such as slopes, bumps and obstacles complex terrain and obstacles. To the best of our knowledge, times and robustness. this last result is the only implementation of a tensegrity robot A static linear feedback controller is designed, which ro- controller demonstrating such capabilities. Figure 12 presents bustly generates a set of desired oscillatory motor signals after such results from within the NTRT simulator. a short learning phase. For this experiment, the target spring- Experimentation demonstrated that the hybrid controller’s cable rest lengths ( ) are generated using a random Matsuoka performance is highly sensitive to some parameters appearing i oscillator [39] (see [38, section 3.1] for oscillator parameters in the CPG equations, such as the ones presented in Eq. 18 and motivation). These signals represent desired actuator sig- and 19. As a result, future work will incorporate other methods nals. We manually scaled target signals so that the resulting to optimize feedback data and compute corrections to more behavior corresponds to a motion pattern with large shape accurately navigate complex environments. A good exam- deformations while keeping the physical structure from mov- ple of such an improvement can be found in Gay et al. [37], ing too fast (as this impeded motion tracking). The resulting where sensory information is preprocessed by a neural network gait was a slow crawling motion that allowed motion-capture and trained using particle-swarm optimization methods before tracking of the full experiment. being fed back to the CPG. In the same idea, Reservoir Com- puting can also be a suitable tool for feedback computation, as The algorithm proceeds by ﬁrst applying target rest lengths detailed in the following section. to actuators in an open-loop setup, inducing the robot to start moving. Next, online learning is applied to approximate de- IV.3 Learning a Matsuoka Oscillator With sired signals based on sensor readings. These approximations are the feedback signals. The ratio of open-loop versus feed- Physical Reservoir Computing back signals is gradually decreased until signals are generated This section presents the ﬁnal controls results of this work: an by the feedback loop alone. At this point, the robot will ro- implementation of the Physical Reservoir Computing (PRC) bustly maintain oscillatory patterns. The precise equations and principle on the ReCTeR hardware prototype. These results val- parameters used in the experiment are provided in [38] and the idate use of the NTRT in an untethered, underactuated feedback supplementary material. control of tensegrity icosahedra with string force sensors [38, In our prior simulation work, we used the term PRC to section 5.1.1]. Here, closed-loop feedback control is used describe how nonlinear computations, which are inherently when motor signals are generated by a Matsuoka oscillator. performed by a physical system, can be easily exploited to This demonstrates a successful adaptation of our simulation simplify control of tensegrities [38]. PRC extends the Reser- results to a physical platform (ReCTeR), with similar learning voir Computing (RC) concept that, at its origin, is a simple

11 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface technique to train recurrent neural networks [40, 41]. The com- VI. Conclusion mon idea is that a system with complex dynamics is perturbed externally, but is otherwise left untouched. Instead, a simple Tensegrity is a curious design concept, spanning art, science, readout mechanism is trained to perform the desired computa- and biology. This work presented the tensegrity workflow de- tional task. A number of related demonstrations have recently veloped at the NASA Ames Research Center. Our simulator se- appeared in the soft robotics fields, e.g., RC applied to a soft, tups were described and demonstrated, and a new, highly com- simulated octopus arm [42, 43]. pliant, untethered tensegrity robot - ReCTeR - was used to vali- Controller feedback signals are obtained from ReCTeR’s 24 date simulator setups in both dynamic and kinematic situations. force transducers. Since these sensors are mounted perpendicu- Next, various control strategies were presented, based on evo- larly to the robot’s struts, output values depend on the angle of lutionary algorithms and CPGs, and a feedback controller was attack and tension of the attached spring-cable assembly. Thus, implemented on the hardware platform to demonstrate sensor the sensors provide a readout of the robot’s state, similar to capabilities. The biologically-inspired control approaches we state observations in RC. The robot’s behavior was evaluated are exploring appear naturally suited for biologically-inspired using the motion capture setup described in section III.1. tensegrity structures, due to their matching nonlinear and oscil- Figure 13 shows the result of an experiment where we first latory qualities. outsourced motor-signal generation to the feedback loop by An important aspect of this work is the creation of an Open online learning of the feedback weights. After training, we Source simulation environment (the NTRT) for tensegrity- disturbed the system (lifting and constraining the robot). In this based mobility and manipulation controls research that has case, the robot stops moving and switches back to its original now been validated against hardware. This simulation environ- oscillatory mode when released, demonstrating robustness of ment enables us to develop an understanding of the structure the learned feedback controller, corresponding to our simula- and qualities of successful control approaches. Using evolution results. tionary exploration of parameters for different structural and This experiment is a first demonstration of a simple, robust biologically-inspired control approaches, this system can be feedback control strategy implemented in both hardware and used to develop performance-driven hardware requirements, simulation for this class of untethered tensegrity robots. Ad- such as the forces experienced in the rods, speed and torque ditionally, this result shows the usefulness of tension sensors requirements for actuators, elasticity constants for springs, and for tensegrity control. These PRC experiments are part of a sensor requirements and placements. Developing the right continuous effort to develop low-level controllers for compli- toolset and design workflow enables progress beyond tenseg- ant robots that maximally exploit the robots’ proper dynamics, rity robots that merely move, and into a realm where tensegrity and that allow mitigation of stringent sensor requirements. We systems purposefully interact with the environment and execute discussed many variations and extensions on the hardware tasks. experiment presented here in our prior simulation work [38]. Acknowledgments

V. Future Work This research was supported by the European Commission’s FP7 program under grant agreement No 248311 - AMARSi Current prototype hardware allows for multiple veriﬁcation and the NASA Innovative Advanced Concepts program. levels of NTRT simulations. However, a more capable robot Ken Caluwaerts was supported by a Ph.D. fellowship of the design is required to implement fully dynamic controls from Research Foundation - Flanders (FWO). Support also came the CPG systems and related work. ReCTeR has a maximum from NSF Graduate Research Fellowship No. DGE1106400, tension-force limit in its cables, as well as with the number and by NASA Prime Contract Number NAS2-03144 awarded of cables actuated. We are currently working on a redesigned to the University of California, Santa Cruz, University Afﬁli- six-strut robot with twice the number of actuators, with torque ated Research Center. and velocity capabilities an order of magnitude higher than The authors would like to thank Ryan Adams, Adrian ReCTeR [44]. This robot will be able to implement the more Agogino, Alice Agogino, Mostafa Ajallooeian, Lee Brownston, advanced control schemes described in this paper. Design of Michiel D’Haene, Stephen R. Ellis, Tom Flemons, Terry Fong, this new robot will also target payload protection, a crucial Jeffrey Friesen, Auke Jan Ijspeert, George Korbel, Stephen feature for space exploration. Levin, Sophie Milam, Kyle Morse, Greg Orzech, In Won Park, On the control side, one of our future goals is further inte- Alexandra Pogue, Brian Tietz, Kagan Tumer, Tim Vets, Tim gration of CPG and RC approaches, to maximally exploit the Waegeman, Kyunam Kim, and the NASA Ames Intelligent advantages of both. Robotics Group.

12 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

external CPG vs. feedback controller fraction end of testing phase

1 600 E

0.5 400

learning, mixed no learning, feedback control 200 0 feedforward/feedback control

vertical position (mm) 0 external CPG fraction 230 232 234 236 238 240 vertical position nodes recovering motion after lifting the robot up

600 600 D

400 400

200 200 A B C D E

vertical position (mm) 0 0 0 50 100 150 200 190 192 194 196 198 200 start learning, mix feedforward and feedback end of learning phase constraining the robot

600 A 600 B 600 C

400 400 400

200 200 200

vertical position (mm) 0 0 0 0 2 4 6 8 80 82 84 86 88 90 120 122 124 126 128 130 time (s) time (s) time (s)

Figure 13: Fast online learning of a static feedback controller for a Matsuoka oscillator on ReCTeR based on uncalibrated strain-gauge sensors. The top left plot shows the fraction of feedforward versus feedback control. During learning, both feedback and feedforward controllers (training signals) are active. The inﬂuence of the open-loop feedforward controller decreases, and when its fraction is below 0.2, learning stops and only the trained feedback controller is active. The left plot on the second row shows vertical coordinates (in mm) of the four end caps with the largest vertical displacements as a function of time. The ﬁve surrounding plots are details of this plot, showing different training and testing phases. A) Fully open-loop control. B) Switch from partially open-loop and feedback control to full feedback control, learning stops. C) We perturb the robot by pushing it down, preventing all movements. D) The feedback controller recovers after the robot was lifted from the ground. E) Behavior of the robot after 250s (170s closed loop).

13 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

controller type main features type of locomotion robot HW/SIM loop actuators sensors references motion capture experiments untethered, underactuated control, and simulator validation rolling, single flop, forward kinematics ReCTeR HW & SIM open 6 (motion capture) Section III reactive controller with coevolution robust feedback controller with minimal assumptions steerable rolling over unknown terrain icosahedron with/without payload SIM closed 24 touch sensors [45] & Section IV.1 Bio-inspired strategies (CPG) robust and bioinspired steerable rolling over unknown terrain icosahedron with payload SIM closed 36 distance sensors Section IV.2 physical reservoir computing robust controller with uncalibrated sensors, link with CPGs crawling (HW), various (SIM) ReCTeR (HW), icosahedron (SIM) HW & SIM closed 6 (HW), variable (SIM) tension sensors [38] & Section IV.3 sine waves with coevolution simple, distributed implementation forward rolling on flat terrain icosahedron with/without payload SIM open 24/36 - [29] stepwise functions with coevolution HW constraints and power consumption information forward rolling icosahedron with/without payload SIM open 24 - Section IV.1 morphological communication communication through body dynamics, distributed control crawling like 15 bar tensegrity tower SIM closed 30 tension sensors [27] genetic algorithms first dynamic locomotion crawling 3 and 4 bar prisms SIM open 9/12 - [1] pneumatic actuators insightful control, original hardware implementation rolling icosahedron with pneumatic actuators HW open 24 - [19, 20] feedback non-linear control control theory approach position/trajectory control any tensegrity SIM closed all full state [15] vibration driven cheap hardware, exploits body dynamics various various HW open 1 - [46] kinematic controllers tested on HW platforms and well studied none various (constrained) HW & SIM both variable - [47, 48] CPG resonance entrainment demonstration of HW CPG control none class two tensegrity beam HW & SIM both 2 linear actuators tension & position [23]

Table 2: Overview of various types of controllers for tensegrity robots and our experiments.

14 Design and Control of Compliant Tensegrity Robots • K. Caluwaerts et al. • J. R. Soc. Interface

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